As students are asked to communicate about the mathematics they are studying--to justify their reasoning to a classmate or to formulate a question about something that is puzzling--they gain insights into their thinking. In order to communicate their thinking to others, students naturally reflect on their learning and organize and consolidate their thinking about mathematics.
Students should be encouraged to increase their ability to express themselves clearly and coherently. As they become older, their styles of argument and dialogue should more closely adhere to established conventions, and students should become more aware of, and responsive to, their audience. The ability to write about mathematics should be particularly nurtured across the grades.
By working on problems with classmates, students also have opportunities to see the perspectives and methods of others. They can learn to understand and evaluate the thinking of others and to build on those ideas. For example, students who try to solve the following problem algebraically may have difficulty setting up the equations:
There are some rabbits and some hutches. If one rabbit is put in each hutch, one rabbit will be left without a hutch. If two rabbits are put in each hutch, one hutch will remain empty. How many rabbits and how many hutches are there?
They may benefit from the insights of students who solve the problem using a visual representation. Students need to learn to weigh the strengths and limitations of different approaches, thus becoming critical thinkers about mathematics (www.nctm.org).
Instructional programs from prekindergarten through grade 12 should enable all students to--
organize and consolidate their mathematical thinking through communication;
communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
analyze and evaluate the mathematical thinking and strategies of others;
use the language of mathematics to express mathematical ideas precisely.
Example: 8th Grade Algebra
Students are asked to explain, using a graph, variations in average temperature in their city vs. the time of year.
In this example, students must use the language of math to explain their findings to their peers and teacher. The students must first cite their dependent and independent variables (average temperature and time of year, respectively) with the emphasis that average temperature depends on the time of year. The students should also discuss the intervals they used for the range of the dependent variable--a range that is too large could mislead the viewer to think that relatively little change occurred in average temperature variation throughout the year. The students may also wish to discuss the slope of the graph at various points of interest and explain that steeper slopes denote a more drastic change in temperature. The students may want to discuss why they did or did not choose to connect the data points on their graphs. All of these instances are examples of students communicating their findings using the language of mathematics.
As students are asked to communicate about the mathematics they are studying--to justify their reasoning to a classmate or to formulate a question about something that is puzzling--they gain insights into their thinking. In order to communicate their thinking to others, students naturally reflect on their learning and organize and consolidate their thinking about mathematics.
Students should be encouraged to increase their ability to express themselves clearly and coherently. As they become older, their styles of argument and dialogue should more closely adhere to established conventions, and students should become more aware of, and responsive to, their audience. The ability to write about mathematics should be particularly nurtured across the grades.
By working on problems with classmates, students also have opportunities to see the perspectives and methods of others. They can learn to understand and evaluate the thinking of others and to build on those ideas. For example, students who try to solve the following problem algebraically may have difficulty setting up the equations:
There are some rabbits and some hutches. If one rabbit is put in each hutch, one rabbit will be left without a hutch. If two rabbits are put in each hutch, one hutch will remain empty. How many rabbits and how many hutches are there?
They may benefit from the insights of students who solve the problem using a visual representation. Students need to learn to weigh the strengths and limitations of different approaches, thus becoming critical thinkers about mathematics (www.nctm.org).
Instructional programs from prekindergarten through grade 12 should enable all students to--
Example: 8th Grade Algebra
Students are asked to explain, using a graph, variations in average temperature in their city vs. the time of year.
In this example, students must use the language of math to explain their findings to their peers and teacher. The students must first cite their dependent and independent variables (average temperature and time of year, respectively) with the emphasis that average temperature depends on the time of year. The students should also discuss the intervals they used for the range of the dependent variable--a range that is too large could mislead the viewer to think that relatively little change occurred in average temperature variation throughout the year. The students may also wish to discuss the slope of the graph at various points of interest and explain that steeper slopes denote a more drastic change in temperature. The students may want to discuss why they did or did not choose to connect the data points on their graphs. All of these instances are examples of students communicating their findings using the language of mathematics.